Differential Equation - Explicit 2-Parameter family of solutions

[jegues]

Homework Statement

Find an explicit two-parameter family of solutions for,

$$\frac{1}{3}y^{''} = 2x\sqrt[3]{(y^{'})^{2}}$$

Homework Equations

The Attempt at a Solution

See figure attached for my attempt at the solution.

I've gotten to the point where I could simply expand out and do termwise integration to obtain my answer but we usually aren't required to anything messy like that in questions like this.

This leads me to think that I've done something wrong, or taken the wrong approach.

Does anyone see any problems/other approaches I should take?

 

[tiny-tim]

hi jegues!

I've gotten to the point where I could simply expand out and do termwise integration to obtain my answer but we usually aren't required to anything messy like that in questions like this.

This leads me to think that I've done something wrong, or taken the wrong approach.

looks ok to me …

except you've misread your own handwriting … it's not 6x³, it's 6x

(and you could have simplified it slightly at the end by changing C/3 to C/2 … C is a dummy constant)

 

[jegues]

hi jegues!


looks ok to me …

except you've misread your own handwriting … it's not 6x³, it's 6x

(and you could have simplified it slightly at the end by changing C/3 to C/2 … C is a dummy constant)

After fixing my mistake, I'm stuck with the integration,

$$\int (x^{2} + \frac{C}{3})^{3}dx$$

Am I forced to expand it out?

 

[tiny-tim]

i think so

(but change C/3 to C)

 

[jegues]

i think so

(but change C/3 to C)

Okay thanks, I just wanted to make sure I wasn't do more work than required.